reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th36:
  for D1,D2 be finite set st D1 misses D2
    for o1 be DoubleReorganization of D1,
        o2 be DoubleReorganization of D2 holds
    o1^o2 is DoubleReorganization  of D1\/D2
proof
  let D1,D2 be finite set such that A1:D1 misses D2;
  let o1 be DoubleReorganization of D1,
      o2 be DoubleReorganization of D2;
  set D=D1\/D2;
  rng o1 c= D*
  proof
    let x be object;
    assume x in rng o1;
    then reconsider x as FinSequence of D1 by FINSEQ_1:def 11;
    rng x c= D1 & D1 c= D by XBOOLE_1:7;
    then rng x c= D;
    then x is FinSequence of D by FINSEQ_1:def 4;
    hence  thesis by FINSEQ_1:def 11;
  end;
  then reconsider O1=o1 as FinSequence of D* by FINSEQ_1:def 4;
  rng o2 c= D*
  proof
    let x be object;
    assume x in rng o2;
    then reconsider x as FinSequence of D2 by FINSEQ_1:def 11;
    rng x c= D2 & D2 c= D by XBOOLE_1:7;
    then rng x c= D;
    then x is FinSequence of D by FINSEQ_1:def 4;
    hence thesis by FINSEQ_1:def 11;
  end;
  then reconsider O2=o2 as FinSequence of D* by FINSEQ_1:def 4;
  A2:Values o1 =D1 & Values o2 =D2 by Def7;
  then A3:O1^O2 is double-one-to-one by A1,Th33;
  Values(O1^O2)=D1\/D2 by A2,Th2;
  hence thesis by A3,Def7;
end;
